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Chapter 8: Problem 10

Solve each equation. $$ \sqrt{5 x-4}=0 $$

Short Answer

Expert verified
x = \( \frac{4}{5} \)

Step by step solution

01

Isolate the Radicand

Identify that the term inside the square root must be zero for the whole expression to equal zero. So, set the expression inside the square root to zero: \[ 5x - 4 = 0 \]
02

Solve for x

Add 4 to both sides of the equation to isolate the term with the variable:\[ 5x = 4 \]Then, divide both sides by 5 to solve for \( x \):\[ x = \frac{4}{5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
To solve square root equations, the first thing you need to do is isolate the variable. This means you want to get the variable alone on one side of the equation. For the given equation \(\text{\sqrt{5x-4} = 0}\), you start by identifying that the term inside the square root, called the radicand, must equal zero. So, you set \(\text{5x - 4 = 0}\). This simplification helps you identify a clear, solvable equation. By isolating the variable, you're making it easier to focus on finding the specific value of \(\text{x}\). When you isolate the radicand, you鈥檙e ensuring precision in the following algebraic steps, thereby applying a fundamental principle of equation solving.
Understanding Square Root Equations
Square root equations involve an expression under a square root symbol. To solve these, first understand that the square root of any non-negative number outputs another non-negative number. Hence, when dealing with an equation like \(\sqrt{5x - 4} = 0\), it鈥檚 crucial to recognize that the only way for the square root to result in zero is if the expression inside it is zero. Therefore, you simplify the equation to \(5x - 4 = 0\). This step is important because it converts the square root equation into a simpler linear equation. Always remember that square root functions have specific properties which must be managed carefully.
Algebraic Manipulation
Algebraic manipulation is essential for solving equations. After isolating the variable, you manipulate the equation to find the variable's value. For instance, from \(5x - 4 = 0\), you add 4 to both sides, resulting in \(5x = 4\). Next, you divide both sides by 5, bringing you to \(x = \frac{4}{5}\). These steps involve basic operations like addition, subtraction, multiplication, and division. The goal is to perform the same operation on both sides of the equation to maintain balance while simplifying the expression to find \(x \). This clear approach ensures that your solution is precise and correctly derived from the given equation.

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